P erturbative Results Without Diagrams R. Rosenfelder - PowerPoint PPT Presentation

P erturbative Results Without Diagrams R. Rosenfelder Paul-Scherrer-Institut, Villigen PSI (Switzerland) PSI, 31 July 2008 arXiv:0805.4525 [hep-th] , submitted to Phys. Rev. E (Computational Physics) 1. Introduction 2. A new method (applied to the polaron g.s. energy) 3. Numerical procedures and results 4. Summary 5. Outlook: application to worldline QED

R. Rosenfelder (PSI) : PT without diagrams 2 1. Introduction Usually in perturbative calculation in Quantum (Field) Theory the number of diagrams grows factorially with the order Example : number of diagrams for g-2 of the electron in QED (see Itzykson & Zuber p. 466, 467) � � 4 z (1 − S ) 1 + K ′ 0 ( z ) , z = − 1 Γ( α ) = , S = − 2 z S 3 K 0 ( z ) 4 α expand in powers of α = 1 / 137 . 036 Γ( α ) = 1 + α + 7 α 2 + 72 α 3 + 891 α 4 + 12672 α 5 + 202770 α 6 + . . . = ⇒ Consequence : huge cancellations between individual diagrams heroic efforts needed for higher-order calculations Schwinger (1948), Petermann, Sommerfield (1957) Laporta & Remiddi (1996), Kinoshita et al. (1990-2005) Need (more modestly: would be nice to have) new methods !

R. Rosenfelder (PSI) : PT without diagrams 3 2. A new method (applied to the polaron g. s. energy) Take as simple (but nontrivial) example the polaron problem – a non-relativistic field theory polaron = electron slowly moving through polarizable crystal model Hamiltonian H. Fr¨ ohlich (1954) a k + √ α � � 1 1 � � p 2 + x + h.c. a † a † k e − i k · ˆ ˆ 2ˆ ˆ k ˆ ˆ H ∼ | k | k k α : dimensionless electron-phonon coupling constant Ground-state energy of polaron: � e n α n E 0 : = , e 1 = − 1 n =1 = − 0 . 01591962 (1959) , e 3 = − 0 . 00080607 Smondyrev (1986) e 2

R. Rosenfelder (PSI) : PT without diagrams 4 In field-theoretic language: have to evaluate self-energy diagrams with more and more loops Long live the PATH INTEGRAL : phonons can be integrated out exactly! Feynman (1955) � β →∞ D 3 x e − S eff e − βE 0 Z ( β ) = − → where for large β � β � β � dt 1 d 3 k 1 x 2 + α dt 1 dt 2 e −| t 1 − t 2 | ⇒ S eff [ x ] ∼ 2 ˙ k 2 exp [ i k · ( x ( t 1 ) − x ( t 2 ))] = : S 0 + S 1 0 0

R. Rosenfelder (PSI) : PT without diagrams 5 Employ cumulant expansion of partition function �� � ( − ) n Z ( β ) = Z 0 exp n ! λ n ( β ) n =1 where λ n ( β ) are the cumulants w.r.t. S 1 m n ( β ) ≡ � S 1 n � ∝ α n Recursion relation with the moments n − 1 � n � � λ n +1 = m n +1 − λ k +1 m n − k k k =0 λ 1 = m 1 m 2 − m 2 λ 2 = 1 m 3 − 3 m 2 m 1 + 2 m 3 λ 3 = 1 m 4 − 4 m 3 m 1 − 3 m 2 2 + 12 m 2 m 2 1 − 6 m 4 = λ 4 1 m 5 − 5 m 4 m 1 − 10 m 3 m 2 + 20 m 3 m 2 1 + 30 m 2 2 m 1 − 60 m 2 m 3 1 + 24 m 5 λ 5 = 1 . . . ( − ) n +1 1 Note : λ n ( β ) ∝ α n = ⇒ e n = lim λ n ( β ) α n n ! β β →∞

R. Rosenfelder (PSI) : PT without diagrams 6 The path integral for the moments � D 3 x S 1 n e − S 0 [ x ] , = m 0 = 1 m n C can be evaluated exactly. Write Coulomb propagator as � ∞ � � k 2 = 1 1 − 1 2 k 2 u du exp 2 0 = ⇒ all momentum integrations can be performed and one obtains � � �� β � t m � ∞ n n � α n � � ( − ) n dt ′ ( t m − t ′ = exp − m ) m n dt m du m m (4 π ) n/ 2 0 0 0 m =1 m =1 · � det A � �� − 3 / 2 t 1 . . . t n , t ′ 1 . . . t ′ n ; u 1 . . . u n with ( n × n )- matrix A � � 1 −| t i − t j | + | t i − t ′ j | + | t ′ i − t j | − | t ′ i − t ′ = j | + u i δ ij A ij 2 ↑ ↑ non-analytic analytic dependence

R. Rosenfelder (PSI) : PT without diagrams 7 Define A ij = : a ij + u i δ ij Diagonal parts: a ii = t i − t ′ i ≡ σ i Non-diagonal matrix elements : 1 � j ) � := t i + t ′ i − ( t j + t ′ S 2 1 r := 2 ( σ i − σ j ) 1 s := 2 ( σ i + σ j )

R. Rosenfelder (PSI) : PT without diagrams 8 3. Numerical procedures and results For m n ( β ) ⇒ λ n ( β ) one has to do a 3n -dimensional integral over t i , t ′ i , u i Two u i -integrations can be done analytically ———————————————————————— � ∞ 2 du n det − 3 / 2 A (1 , 2 , . . . , n ) = n � A n det n A ( u n = 0) 0 arcsin √ x HF � ∞ � ∞ 4 du n det − 3 / 2 du n − 1 A (1 , 2 , . . . , n ) = √ x HF n � A n − 1 ,n A n − 1 A n 0 0 where A n , A n − 1 , A n − 1 ,n are principal minors of the determinant det n A ≡ A 0 ≤ x HF : = 1 − A n − 1 ,n A ≤ 1 A n − 1 A n because A ij is a positive semi-definite matrix = ⇒ Hadamard-Fischer inequality A n − 1 A n ≥ A n − 1 ,n A ———————————————————————— After performing u n , u n − 1 -integrations analytically = ⇒ ( 3n − 2 )-dimensional integral left

R. Rosenfelder (PSI) : PT without diagrams 9 Useful trick : calculate directly ∂λ n ∂β = ⇒ ( 3n − 3 )-dimensional integral ! Further advantage: asymptotic behaviour (thus extrapolation to β → ∞ ) is much improved: � β →∞ e n ( β ) : = ( − ) n +1 � ∂λ n ∂ → e n − a n √ β e − β (?) + . . . × − . . . = β · e n + const − α n n ! ∂β ∂β Exponential convergence to e n : analytically proved for n = 1 , 2 numerically for n = 3 : assume e n ( β ) → e n − a n β − κ n e − β fit to Monte-Carlo data gives κ 3 = 0 . 55(3) Assume it also for n > 3 ...

R. Rosenfelder (PSI) : PT without diagrams 10 Numerical evaluation: mapping to hypercube [0 , 1] , then: Monte-Carlo integration with VEGAS program or routines from the CUBA library Note : Monte-Carlo integration can handle non-analytic, even discontinous integrands

R. Rosenfelder (PSI) : PT without diagrams 11 check n = 3 (6-dimensional integral): analytical: e 3 = − 0 . 80607005 · 10 − 3

R. Rosenfelder (PSI) : PT without diagrams 12 but for n = 4 convergence is slow with number of function calls: solution: perform the ( n − 2) remaining u i -integrations by deterministic integration routine. Very efficient: tanh-sinh-method !

R. Rosenfelder (PSI) : PT without diagrams 13 − → Transformation x = g ( t ) = tanh ( sinh t ) t ∈ [ −∞ , + ∞ ] cosh t g ′ ( t ) = cosh 2 (sinh t ) Euler − MacLaurin = ⇒ � +1 � + ∞ k =+ ∞ � dx f ( x ) = dt g ′ ( t ) f ( g ( t )) ≈ h w k f ( x k ) − 1 −∞ k = −∞ with x k = g ( kh ) , w k = g ′ ( kh ) w k double exponentially decreasing for large | k |

R. Rosenfelder (PSI) : PT without diagrams 14 now for n = 4:

R. Rosenfelder (PSI) : PT without diagrams 15 and also n = 5 (evaluation of (9+3)-dim. integral) is within reach: Here tanh-sinh-integration seems to be slightly better than Gauss-Legendre

R. Rosenfelder (PSI) : PT without diagrams 16

R. Rosenfelder (PSI) : PT without diagrams 17 4. Summary • Two additional perturbative coefficients e 4 , e 5 for the polaron g.s. energy have been determined by a new (mostly) numerical method. This amounts to performing a 4-loop and 5-loop calculation in Quantum Field Theory • Method is based on a combination of Monte-Carlo integration techniques and deter- ministic quadrature rules for finite β (temperature) and on judicious extrapolation to β → ∞ (zero temperature). Reproduces Smondyrev’s coefficient e 3 with high accuracy • Cancellation in n th order not among many individual diagrams but among much fewer terms in the integrand of the (3 n − 3)-dimensional integral • Increased computational power would allow to improve accuracy for e 4 , e 5 and even e 6 seems accessible • Application to g-2 of the electron under investigation (worldline representation of QED). Challenge: renormalization !?

R. Rosenfelder (PSI) : PT without diagrams 18 5. Outlook: application to worldline QED Generating functional � � � D ¯ i S [ ¯ ψ, ψ, A ] + ( ¯ Z [¯ η, η, j ] = ψ D ψ D A exp ψ, η ) + (¯ η, ψ ) + ( j, A ) � � � � S [ ¯ ¯ ψ, ψ, A ] = γ · ( i∂ − eA ) − M 0 + S 0 [ A ] ψ, ψ � �� � ≡ Π 2-point function: � � � � � � � 1 � ψ (0) � e iS 0 [ A ] Det ( γ · Π − M 0 ) � � ψ ( x ) ¯ ∼ D A x 0 � � γ · Π − M 0 � �� � � � � �� � ↓ ↑ = const. in quenched approx. Schwinger trick ↓ � ∞ � � � dχ exp � i � � T − i ( γ · Π + M 0 ) χ � ( γ · Π) 2 − M 2 ∼ dT dχ = 0 , dχ χ = 1 0 0 Berezin

R. Rosenfelder (PSI) : PT without diagrams 19 result in momentum space : ( Alexandrou, RR & Schreiber, PR A 59 (1999)) � ∞ � � � � � D 4 ζ e i S [ x,ζ,χ,A ] � � dχ e − i ( M 2 0 T + M 0 χ ) d 4 x e − ip · x D A e iS 0 [ A ] D 4 x G 2 ( p ) ∼ dT � Γ → γ 0 orbital trajectory : x (0) = 0 , x ( T ) = x spin trajectory : ζ (0) + ζ ( T ) = Γ � T � � − 1 x 2 + iζ · ˙ S [ x, ζ, χ, A ] ∼ dt 2 ˙ ζ + ˙ x · ζ χ − e ˙ x · A − ie ζ · F · ζ 0 ↑ ↑ ↑ spin-orbit convection spin current Photon field A can be integrated out exactly = ⇒ effective action